Clique-factors in graphs with sublinear $\ell$-independence number

TL;DR

This paper determines the precise minimum degree conditions needed for the existence of clique-factors in large graphs with sublinear $ ext{l}$-independence number, extending classical results to a new setting.

Contribution

It establishes the asymptotically sharp minimum degree threshold for $K_r$-factors in graphs with $ ext{l}$-independence number $n^{1-o(1)}$, addressing a recent open question.

Findings

Identifies the exact asymptotic minimum degree threshold for clique-factors.

Extends Hajnal--Szemerédi theorem to graphs with sublinear $ ext{l}$-independence number.

Provides a Ramsey--Turán type result for clique-factors in this setting.

Abstract

Given a graph $G$ and an integer $\ell\ge 2$, we denote by $\alpha_{\ell}(G)$ the maximum size of a $K_{\ell}$-free subset of vertices in $V(G)$. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$-vertex graphs $G$ with $\alpha_{\ell}(G) = o(n)$, which can be seen as a Ramsey--Tur\'an variant of the celebrated Hajnal--Szemer\'edi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$-factors in $n$-vertex graphs $G$ with $\alpha_\ell(G)=n^{1-o(1)}$ for all $r\ge \ell\ge 2$.